<h2>Problem 29</h2>
<div style="color:#666;font-size:80%;">25 October 2002</div><br />
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<p>Consider all integer combinations of <i>a</i><img src="" style="display:none;" alt="^(" /><sup><i>b</i></sup><img src="" style="display:none;" alt=")" /> for 2 <img src='images/symbol_le.gif' width='10' height='12' alt='&le;' border='0' style='vertical-align:middle;' /> <i>a</i> <img src='images/symbol_le.gif' width='10' height='12' alt='&le;' border='0' style='vertical-align:middle;' /> 5 and 2 <img src='images/symbol_le.gif' width='10' height='12' alt='&le;' border='0' style='vertical-align:middle;' /> <i>b</i> <img src='images/symbol_le.gif' width='10' height='12' alt='&le;' border='0' style='vertical-align:middle;' /> 5:</p>
<blockquote>2<img src="" style="display:none;" alt="^(" /><sup>2</sup><img src="" style="display:none;" alt=")" />=4, 2<img src="" style="display:none;" alt="^(" /><sup>3</sup><img src="" style="display:none;" alt=")" />=8, 2<img src="" style="display:none;" alt="^(" /><sup>4</sup><img src="" style="display:none;" alt=")" />=16, 2<img src="" style="display:none;" alt="^(" /><sup>5</sup><img src="" style="display:none;" alt=")" />=32<br />
3<img src="" style="display:none;" alt="^(" /><sup>2</sup><img src="" style="display:none;" alt=")" />=9, 3<img src="" style="display:none;" alt="^(" /><sup>3</sup><img src="" style="display:none;" alt=")" />=27, 3<img src="" style="display:none;" alt="^(" /><sup>4</sup><img src="" style="display:none;" alt=")" />=81, 3<img src="" style="display:none;" alt="^(" /><sup>5</sup><img src="" style="display:none;" alt=")" />=243<br />
4<img src="" style="display:none;" alt="^(" /><sup>2</sup><img src="" style="display:none;" alt=")" />=16, 4<img src="" style="display:none;" alt="^(" /><sup>3</sup><img src="" style="display:none;" alt=")" />=64, 4<img src="" style="display:none;" alt="^(" /><sup>4</sup><img src="" style="display:none;" alt=")" />=256, 4<img src="" style="display:none;" alt="^(" /><sup>5</sup><img src="" style="display:none;" alt=")" />=1024<br />
5<img src="" style="display:none;" alt="^(" /><sup>2</sup><img src="" style="display:none;" alt=")" />=25, 5<img src="" style="display:none;" alt="^(" /><sup>3</sup><img src="" style="display:none;" alt=")" />=125, 5<img src="" style="display:none;" alt="^(" /><sup>4</sup><img src="" style="display:none;" alt=")" />=625, 5<img src="" style="display:none;" alt="^(" /><sup>5</sup><img src="" style="display:none;" alt=")" />=3125<br /></blockquote>
<p>If they are then placed in numerical order, with any repeats removed, we get the following sequence of 15 distinct terms:</p>
<p style='text-align:center;'>4, 8, 9, 16, 25, 27, 32, 64, 81, 125, 243, 256, 625, 1024, 3125</p>
<p>How many distinct terms are in the sequence generated by <i>a</i><img src="" style="display:none;" alt="^(" /><sup><i>b</i></sup><img src="" style="display:none;" alt=")" /> for 2 <img src='images/symbol_le.gif' width='10' height='12' alt='&le;' border='0' style='vertical-align:middle;' /> <i>a</i> <img src='images/symbol_le.gif' width='10' height='12' alt='&le;' border='0' style='vertical-align:middle;' /> 100 and 2 <img src='images/symbol_le.gif' width='10' height='12' alt='&le;' border='0' style='vertical-align:middle;' /> <i>b</i> <img src='images/symbol_le.gif' width='10' height='12' alt='&le;' border='0' style='vertical-align:middle;' /> 100?</p>

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